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No Intercept Regression and Analysis of Variance

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Example Data Set YX 520 623 727 833 831 935 1043 519 625 729 831

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Estimate two models Model with y-intercept Y = a + b * X Regression Statistics Multiple R0.984 R Square0.969 Adjusted R Square0.965 Standard Error0.300 Observations11 Model no y-intercept Y = b * X Regression Statistics Multiple R0.999 R Square0.998 Adjusted R Square0.898 Standard Error0.333 Observations11

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Observations The model with a y-intercept is more complex than the model with no y-intercept. One would expect then that the R 2 of the model would decline when the y-intercept is removed. BUT, the R 2 actually increases. If the explanatory power of the model, R 2, increases, then the error of the model, Standard Error, should decrease. But, the Standard Error actually increases.

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Analysis of Variance Table model with y-intercept ANOVA dfSSMSFSignificance F Regression124.83 276.720.000000 Residual90.810.09 Total1025.64

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Analysis of Variance Table model no y-intercept ANOVA dfSSMSFSignificance F Regression1591.89 5338.740.000000 Residual101.110.11 Total11593.00

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Comparison of Sum of Squares Y-intercept dfSS Regression124.83 Residual90.81 Total1025.64 No y- intercept dfSS Regression1591.89 Residual101.11 Total11593.00

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Revision of Sum of Squares for no-intercept model No y-interceptdfSS Regression1 SST – SSE 25.64 – 1.11 = 24.53 Residual 9 1.11 Total 1025.64 These are from the model with a y-intercept. This is re-calculated.

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Comparison of Revised Sum of Squares Y-intercept dfSS Regression124.83 Residual90.81 Total1025.64 No y- intercept dfSS Regression124.53 Residual91.11 Total1025.64

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Revised Statistics Model no y-intercept Y = b * X Regression Statistics Multiple R0.999 R Square0.998 Adjusted R Square0.898 Standard Error0.333 Observations11 SSR / SST = 24.53 / 25.64 = 0.957 √ SSE / d.o.f. = √ 1.11 / 9 = 0.351

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Comparison of Revised Statistics Model with y-intercept Y = a + b * X Regression Statistics Multiple R0.984 R Square0.969 Adjusted R Square0.965 Standard Error0.300 Observations11 Model no y-intercept Y = b * X Regression Statistics Multiple R0.999 R Square0.957 Adjusted R Square0.898 Standard Error0.351 Observations11

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Revised Observations The model with a y-intercept is more complex than the model with no y-intercept. One would expect then that the R 2 of the model would decline when the y-intercept is removed. BUT, the R 2 actually increases. If the explanatory power of the model, R 2, increases, then the error of the model, Standard Error, should decrease. But, the Standard Error actually increases.

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Analysis of Variance Table model no y-intercept REVISED ANOVA dfSSMSFSignificance F Regression124.53 199.430.000000 Residual91.110.123 Total1025.64 SS / df MSR/ MSE FDIST (199.43, 1,9)

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Summary ANOVA dfSSMSFSignificance F Regression1YYYY ResidualX1.11Y TotalXX When comparing a model with a y-intercept to the same model without a y-intercept 1.Revise the ANOVA table for the no-intercept model with values from the y-intercept model (X). 2.Recalculate necessary items (Y), and the R 2 and the Standard Error.

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